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Antti H S Laaksonen 2017-04-21 22:18:22 +03:00
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chapter25.tex
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@ -26,13 +26,13 @@ On each move, the player has to remove
and the player who removes the last stick wins the game. and the player who removes the last stick wins the game.
For example, if $n=10$, the game may proceed as follows: For example, if $n=10$, the game may proceed as follows:
\begin{enumerate}[noitemsep] \begin{itemize}[noitemsep]
\item Player $A$ removes 2 sticks (8 sticks left). \item Player $A$ removes 2 sticks (8 sticks left).
\item Player $B$ removes 3 sticks (5 sticks left). \item Player $B$ removes 3 sticks (5 sticks left).
\item Player $A$ removes 1 stick (4 sticks left). \item Player $A$ removes 1 stick (4 sticks left).
\item Player $B$ removes 2 sticks (2 sticks left). \item Player $B$ removes 2 sticks (2 sticks left).
\item Player $A$ removes 2 sticks and wins. \item Player $A$ removes 2 sticks and wins.
\end{enumerate} \end{itemize}
This game consists of states $0,1,2,\ldots,n$, This game consists of states $0,1,2,\ldots,n$,
where the number of the state corresponds to where the number of the state corresponds to
@ -66,7 +66,7 @@ is a winning state for the opponent.
More generally, if there is a move that leads More generally, if there is a move that leads
from the current state to a losing state, from the current state to a losing state,
the current state is a winning state, the current state is a winning state,
and otherwise it is a losing state. and otherwise the current state is a losing state.
Using this observation, we can classify all states Using this observation, we can classify all states
of a game starting with losing states where of a game starting with losing states where
there are no possible moves. there are no possible moves.
@ -118,7 +118,7 @@ can be classified as follows
It is easy to analyze this game: It is easy to analyze this game:
a state $k$ is a losing state if $k$ is a state $k$ is a losing state if $k$ is
divisible by 4, and otherwise it divisible by 4, and otherwise it
is winning state. is a winning state.
An optimal way to play the game is An optimal way to play the game is
to always choose a move after which to always choose a move after which
the number of sticks in the heap the number of sticks in the heap
@ -247,7 +247,7 @@ and this is always the final state.
It turns out that we can easily classify It turns out that we can easily classify
any nim state by calculating any nim state by calculating
the \key{nim sum} $x_1 \oplus x_2 \oplus \cdots \oplus x_n$, the \key{nim sum} $s = x_1 \oplus x_2 \oplus \cdots \oplus x_n$,
where $\oplus$ is the xor operation\footnote{The optimal strategy where $\oplus$ is the xor operation\footnote{The optimal strategy
for nim was published in 1901 by C. L. Bouton \cite{bou01}.}. for nim was published in 1901 by C. L. Bouton \cite{bou01}.}.
The states whose nim sum is 0 are losing states, The states whose nim sum is 0 are losing states,
@ -312,11 +312,11 @@ one bit in the nim sum:
\begin{center} \begin{center}
\begin{tabular}{r|r} \begin{tabular}{r|r}
10 & \texttt{10\textcircled{1}0} \\ 10 & \texttt{10\underline{1}0} \\
12 & \texttt{1100} \\ 12 & \texttt{1100} \\
5 & \texttt{0101} \\ 5 & \texttt{0101} \\
\hline \hline
3 & \texttt{00\textcircled{1}1} \\ 3 & \texttt{00\underline{1}1} \\
\end{tabular} \end{tabular}
\end{center} \end{center}
@ -340,7 +340,8 @@ which is a losing state:
\index{misère game} \index{misère game}
In a \key{misère game}, the goal is opposite, In a \key{misère game}, the goal of the game
is opposite,
so the player who removes the last stick so the player who removes the last stick
loses the game. loses the game.
It turns out that a misère nim game can be It turns out that a misère nim game can be
@ -368,8 +369,8 @@ so the nim sum is not 0.
\index{Sprague–Grundy theorem} \index{Sprague–Grundy theorem}
The \key{Sprague–Grundy theorem}\footnote{The theorem was discovered The \key{Sprague–Grundy theorem}\footnote{The theorem was
independently by R. Sprague \cite{spr35} and P. M. Grundy \cite{gru39}.} generalizes the independently discovered by R. Sprague \cite{spr35} and P. M. Grundy \cite{gru39}.} generalizes the
strategy used in nim to all games that fulfil strategy used in nim to all games that fulfil
the following requirements: the following requirements:
@ -755,7 +756,7 @@ The Grundy numbers for the mazes are as follows:
In the initial state, the nim sum of the Grundy numbers In the initial state, the nim sum of the Grundy numbers
is $2 \oplus 3 \oplus 3 = 2$, so is $2 \oplus 3 \oplus 3 = 2$, so
the first player can win the game. the first player can win the game.
An optimal move is to move two steps up One optimal move is to move two steps up
in the first maze, which produces the nim sum in the first maze, which produces the nim sum
$0 \oplus 3 \oplus 3 = 0$. $0 \oplus 3 \oplus 3 = 0$.